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(hyperbolic) symmetry groups 
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#1
Sep2810, 08:27 AM

P: 625

Hello,
how do symmetry groups in the Euclidean space differ from the symmetry groups in the hyperbolic space (in the Poincaré disk) ? I've been told that in the hyperbolic case one has at disposal a richer "vocabulary" to describe symmetries, but I don't see how, and maybe I misunderstood. Can anyone give me a hint and/or some source? Thanks 


#2
Sep2810, 09:29 AM

P: 1,411

You may like to start with Hyperbolic tesselations



#3
Sep2810, 10:06 AM

P: 625

Now I am wondering if in the hyperbolic case we have the same families of symmetries like Reflections, Rotations and so on...or if we can actually define something new.
I'm just trying to figure out where is this "extra richness" in the hyperbolic case that we cannot find in the Euclidean plane. 


#4
Sep2810, 10:34 AM

P: 1,411

(hyperbolic) symmetry groups
Euclidean plane, as a Riemannian manifold, is flat. In the hyperbolic case you have different kinds of rotations and reflections.



#5
Sep2810, 10:37 AM

P: 625

If yes, could you mention one, or some? Thanks! 


#6
Sep2810, 10:39 AM

P: 1,411

Think of a scalar product of signature (p,q). You will have subspaces with all kind of signatures. Each kind gives you a different submanifold of the corresponding Grassmann manifold. A whole ZOO.



#7
Sep2810, 10:44 AM

P: 1,411

You will have noncompact subgroups and the interplay between compact and noncompact.
The richness comes, for instance in the natural compactification. In the Euclidean case [tex]R^n[/tex] has a natural onepoint compactification. In the hyperbolic case the "conformal infinity" has some interesting structure. 


#8
Sep2810, 03:52 PM

PF Gold
P: 2,432

In the euclidean plane, the tesselations are all the same size. In the hyperbolic plane, they are of every size  symmetric around a powerlaw axis of scale. http://www.mcescher.com/Gallery/recognbmp/LW436.jpg 


#9
Sep2910, 02:08 AM

P: 625

I find it difficult to spot this extra symmetry from the pictures, especially from Escher's pictures. It seems to me that all these illustrations exhibit an ordinary rotational symmetry. In the case of the picture you mentioned 3fold symmetry.
I guess I am wrong, but I dont know where. 


#10
Sep2910, 03:28 AM

P: 1,411

You also have scaling symmetry. Look at that:
x^3y+y3^z+z^3x = 0 Can there be an equation more simple than that? It describes „The Riemann Surface of Klein with 168 Automorphisms”. And yet this simple equation, when analyzed, gives rise to beautiful 168 triangles representing "the fundamental domain". The method of drawing is extremely simple. ;) It is described in a paper "The Riemann Surface of Klein with 168 Automorphisms" by HARRY E. RAUCH1 AND J. LEWITTES Research partially sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. AF691641 .... ….We call attention to an incorrect answer to (iii) by Hurwitz ([7], p. 159, criticized in [1]) and an abortive attempt on (i), (ii), and (iii) by Poincare in [16], p. 130, all noticed after the completion of our work.2 2. Klein's surface Klein originally obtained his surface S in the form of the upper halfplane identified under the principal congruence subgroup of level seven, Gamma(7), of the modular group Gamma. In this form it is necessary to compactify the fundamental domain at its cusps. Klein's group then appears as Gamma/Gamma(7), which is simple and of order 168. We need, however, another representation given by Klein, one which we recognize today as the unit circle uniformization of S. In the unit circle draw the vertical diameter L1 and another diameter L3 making an angle of Pi/7 with L1 and going down to the right. In the lower semicircle draw the arc L2 of the circle which is orthogonal to the unit circle and to L1 and which meets L3 at the angle Pi/3. Let t be the nonEuclidean triangle enclosed by L1,L2,L3 and let Ru R2i R3 be the nonEuclidean reflections in L1,L2,L3respectively. R1,R2,R3 generate a nonEuclidean crystallographic group, which we denote by (2, 3, 7)', with t as a fundamental domain. The images of t under (2, 3, 7)' are a set of nonEuclidean triangles each of which is congruent or symmetric to t according as the group element which maps t on it has an even or odd number of letters as a word in R1 R2i R3. These triangles form a nonEuclidean plastering or tesselation of the interior of the unit circle. The union of t and its image under R2 is a fundamental domain (with suitable conventions about edges) for the triangle group (2, 3, 7), which is the group generated by ….. A convenient fundamental domain for N is the circular arc (nonEuclidean) 14gon Delta shown in Fig. 1. It will be noticed that t appears as the unshaded triangle immediately below and to the right of P0. There are 168 unshaded triangles, which are the images of t under Gamma168 in Delta and 168 shaded triangles, which are the images of t under anticonformal elements of (2, 3, 7)'. …. And here, attached, is my own rendering: 


#11
Sep2910, 04:59 AM

P: 625

Thanks a lot arkajad and apeiron!
Now I start to see better what's going on, although I admit the topic seems to be much deeper than I initially thought, and I need to spend some time to analyze it better. At least, it is now clear that in the hyperbolic case (e.g. in the Poincaré disk) we can see a sort of "scalesymmetry" because the tessellations in the hyperbolic plane are more "exotic" than in the Euclidean plane. @arkajad: I seem to have difficulties finding an electronic version of that paper. Thanks a lot! 


#12
Sep2910, 06:25 AM

P: 1,411




#13
Sep2910, 06:29 AM

P: 625

I still have to study more basic stuff, then if I really need to access that paper I might ask you again. Thanks. See you. 


#14
Sep2910, 06:42 AM

P: 1,411

@mnb96
Alan F. Beardon, "Algebra and geometry" has a nice chapter 15  "Hyperbolic geometry". I think you may like it. 14 Group actions 284 14.1 Groups of permutations 284 14.2 Symmetries of a regular polyhedron 290 14.3 Finite rotation groups in space 295 14.4 Groups of isometries of the plane 297 14.5 Group actions 303 15 Hyperbolic geometry 307 15.1 The hyperbolic plane 307 15.2 The hyperbolic distance 310 15.3 Hyperbolic circles 313 15.4 Hyperbolic trigonometry 315 15.5 Hyperbolic threedimensional space 317 15.6 Finite Mobius groups 319 


#15
Sep2910, 07:51 AM

P: 625

I got the book you suggested from the library.
It seems well written and will be useful. Thanks! 


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